Exit problems for the difference of a compound Poisson process and a compound renewal process

Abstract

In this paper we solve a two-sided exit problem for a difference of a compound Poisson process and a compound renewal process. More specifically, we determine the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this time instant. The results obtained are applied to solve the two-sided exit problem for a particular class of stochastic processes, i.e. the difference of the compound Poisson process and the renewal process whose jumps are exponentially distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. We determine the Laplace transforms of the busy period of the systems M-x vertical bar G(delta)vertical bar 1 vertical bar B, G(delta)

vertical bar 1 vertical bar B in case when delta similar to exp(lambda). Additionally, we prove the weak convergence of the two-boundary characteristics of the process to the corresponding functionals of the standard Wiener process.